This question is motivated by the previous MO question: Show that $ ( sum_ {k = 1} ^ {n} x_ {k} cos {k}) ^ 2 + ( sum_ {k = 1} ^ {n} x_ {k} sin {k}) ^ 2 le (2+ frac {n} {4}) sum_ {k = 1} ^ {n} x ^ 2_ {k} $.

It is a clean asymptotic version of that question.

Leave $ f $ be a non-negative, periodic function with period $ 1 $and square integrable in $ { Bbb R} / { Bbb Z} $. It is true that

$$

| { widehat f} (1) | ^ 2 = Big | int_0 ^ 1 f (x) e ^ {- 2 pi ix} dx Big | ^ 2 le frac 14 int_0 ^ 1 f (x) ^ 2 dx ?

$$

Equality is achieved, for example, when $ f (x) = max (0, cos (2 pi x)) $.

Note that $ | widehat f (1) | = | widehat f (-1) | $ and from $ f $ it is not negative $ | widehat f (1) | le widehat f (0) $. Thus

$$

int_0 ^ 1 f (x) ^ 2 dx = sum_n | widehat f (n) | ^ 2 ge 3 | widehat f (1) | ^ 2,

$$

so that the estimate is maintained with $ 1/3 $ instead of $ 1/4 $. There is a lot of room to improve this argument, and with a more careful application of Bessel's inequality I could get the constant $ 1/4 + 1/4 pi $. But the alleged inequality looks very clean, and I wonder if (i) is true!, (Ii) is known in some context, and (iii) (hopefully) has an elegant proof?